My HTC one 2013 won't start after battery running low. Screen is blank, and no led. Tried charging up by usb and mains, phone is still dead.

I haven't dropped it or damaged it in any way, it just went into sleep mode after battery was low and wont start up again.

Got loads of content on there that I really need, any ideas?

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It sounds like the battery is completely dead. You might need to send it off to HTC to have the battery replaced. You also might be able to do it yourself, just check out YouTube for a video on how to replace the battery.